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Weird Mathematics And The Planets

Ian Beardsley

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It is as if the planets interior to the asteroid belt are distributed by doing what I call weird

calculus. And, that the planets exterior to the asteroid belt are doing normal calculus. It is as if

the planets interior to the asteroid belt are trying to take the derivative of x to the n without

using logarithms. There seems to be a connection between artiﬁcial intelligence elements and

the planets.!

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Table of Contents

Quantization of Planetary Orbits In Terms of AI Binary……….5"

The Conundrum……………………………………………………7"

The Weird Calculus………………………………………………….8

Understanding The Weird Calculus………………………………….9

The Golden Ratio…………………………………………………….12

Weird Arithmetic……………………………………………………..15

Mathematical Look At Weird Calculus……………………………….24

Silicon and Carbon……………………………………………………32

Germanium and Carbon………………………………………………33

The Scheme…………………………………………………………..35

The Planets and AI……………………………………………………40

The Protoplanetary Disc………………………………………………52

Bone…………………………………………………………………..55

Boron………………………………………………………………….67

Masculine And Feminine……………………………………………..85

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Quantization of Planetary Orbits In Terms of AI Binary

I have devised a scheme for the planets in terms of the golden ratio conjugate phi and Euler’s

number e:"

So that for the planets exterior to the asteroid belt or where

is is the asteroid belt ( )

Which is the solution to the differential equation

Where have we seen this? In computer science.

means

Where n is the number of bits in a number N in binary. We write in binary

0=0

1=1

10=2

11=3

100=4

101=5

110=6

(1 − ϕ)e

= 0.7AU = Venus

ϕe

(1−ϕ)

= 0.9AU = Ear th

(2−ϕ)

= 1.52 = M ars

2ϕe

(2−ϕ)

= 4.9 = Jupiter

4ϕe

(2−ϕ)

= 10 = Sat ur n

8ϕe

(2−ϕ)

= 19.69 = Uranus

16ϕe

(2−ϕ)

= 39.38 = Neptune

= 2

ϕe

(2−ϕ)

= c2

c = ϕe

(2−ϕ)

= 2.461

C 2

= C = 2.461

= Jupiter

= Sat ur n

= Ura nu s

= Nept u ne

− 2log(2)

+ log

(2)y = 0

log

N = n

= N

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111=7

1000=8

1001=9

1010=10

1011=11

1100=12

1101=13

1110=14

1111=15

10000=16…

But what is interesting about this?

You can’t have a fractional number of bits, thus the spectrum is quantized according to whole

number solutions of

But so are the planets given by

Meaning, since we have 2, 4, 8. 16 that the planets are quantized into whole number orbits

according to computer binary with Jupiter as 2, Saturn as 4, Uranus as 8, and Neptune as 16 if

we do it in terms of Euler’s number, e and the golden ratio conjugate, ."

That is, 2=10, 4=100, 8=1000, 16=10000"

Are all zeros after a one.!

log

3 = n

n =

log3

log2

= 1.5847

= N

= c2

2ϕe

(2−ϕ)

= 4.9 = Jupiter

4ϕe

(2−ϕ)

= 10 = Sat ur n

8ϕe

(2−ϕ)

= 19.69 = Uranus

16ϕe

(2−ϕ)

= 39.38 = Neptune

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The Conundrum

It is as if the planets interior to the asteroid belt are distributed by doing what I call weird

calculus. And, that the planets exterior to the asteroid belt are doing normal calculus. It is as if

the planets interior to the asteroid belt are trying to take the derivative of x to the n without

using logarithms. This in the sense that:"

If we refer back to the foundations of calculus, while the integral of simple functions can be

considered

We have a conundrum for

That the power rule gives:

Thus to get around this, we searched for a function such that the integral holds, and as such we

discovered the natural logarithm (ln) and Euler’s number e. And we have

Where

And, the derivative of is itself and e is the transcendental and irrational number given by

e=2.718…

That is, while

∫

d x =

n+1

n + 1

+ C

f (x) =

= x

−1

∫

d x =

−1+1

∫

d x = ln(x) + C

= e

ln(x) = log

(x)

−1

ln(x) ≠

ln(x)

−1

ln(x) = e

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We can approximate any function with a polynomial, the simplest example being the linear

approximation formed by writing the change in f(x) due to a change in x:

This results in Taylor’s formula

From which we derive the Taylor series

We know the kth derivative of e to the x is e to the x itself. Thus,

The Weird Calculus

The planets seem to the think

But, rather is

f (x) = f (a) + f′ (a)(x − a)

f (x) = f (a) + f′ (a)(x − a) +

f′ ′ (a)

(x − a)

+ …

∞

∑

n=0

(a)

(x − a)

= f (a) + f′ (a)(x − a) +

f′ ′ (a)

(x − a)

+ …

(k)

(x) = e

= 1 + x +

+ …

lim

n→∞

= 0

∞

∑

n=0

= 1 + x +

+ …

e =

∞

∑

n=0

= 1 +

+ … = 2.718

≠ e

= xe

−(x−1)

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makes me think of the derivative of x to the n:"

And so does"

Because we have"

yields"

by way of"

= "

And, the is like ."

And,…"

makes me think of the second derivative of x to the n:"

That is:"

Because we have"

Thus, in general the derivative in weird calculus is:"

Understanding The Weird Calculus

The plot of has got to be one of the most interesting things I have ever seen:"

(1 − ϕ)e

= n x

(n−1)

ϕe

(1−ϕ)

(n − 1)

(1 − ϕ)

−(ϕ − 1)

(1 − ϕ)

ϕe

n x

(2−ϕ)

= n(n − 1)x

(n−2)

d x

n x

(n−1)

= n(n − 1)x

(n−1−1)

= n(n − 1)x

(n−2)

(n − 2)

−(ϕ − 2) = (2 − ϕ)

(k)

= n

−(n−k)

n x

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Thus in regular calculus"

And, in weird calculus"

This gives"

Let us compare regular calculus to weird calculus:"

If we take"

1.10428"

Where"

and has the property ."

But if we use weird calculus to take the second derivative (written respect to x for weird

derivative)"

It is exactly the Mars orbit."

And, if we simply take, we get…"

=8.15956"

= e

= xe

−(x−1)

dϕ

= ϕe

−(ϕ−1)

= 0.9055

dϕ

= e

= 1.855

1.855

0.9055

= 2.04859 ≈ 2

dϕ

ϕe

(ϕ−1)

= e

(ϕ−1)

(ϕ + 1) = e

(ϕ−1)

1.618

Φ =

Φ = ϕ + 1

= ϕ

−(ϕ−2)

= 1.52

dϕ

ϕe

ϕ+1

= e

ϕ+1

(ϕ + 1) = e

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I think the planets do weird calculus because it is a doubling eﬀect in that"

Because it keeps the planets from interfering with one another so they don’t get torn apart as

they did with the asteroid belt. The second derivative of e to the phi is itself so the 1.855 is

constant. Comparing this to the second derivative of weird calculus we have:"

Notice that"

The Golden Ratio

But what is the golden ratio and its conjugate ."

We can derive the golden ratio as such (refer to ﬁg 13):"

1.855

0.9055

= 2.04859 ≈ 2

1.855

1.52

= 1.220

1.22

= 0.61 ≈ ϕ

= Φ

a = b + c

ac = b

c = a − b

a(a − b) = b

− ab − b

= 0

−

− 1 = 0

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Let us say a/b=x, the golden ratio. Then,…"

Let us diﬀerentiate this implicitly:"

Which is similar to Euler’s number, e because it is the base such that is itself :"

But "

−

= 1 +

(

)

−

= 1 +

(

)

−

(

−

)

−

Φ =

5 + 1

ϕ =

5 − 1

− x − 1 = 0

−

x −

1 = 0

2x − 1 = 0

x =

= e

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Which says for this angle the x-component equals the y-component is that is , x=1/2

bisects a right angle. Which similar in concept to Euler’s number e because it is the base such

that is itself . But if , then:"

It is the diagonal of the unit square. We notice something interesting happens:"

, , "

Where is the cosine of 30 degrees, in the unit equilateral triangle in which the altitude has

been drawn in (ﬁg 14):"

sin 45

∘

= cos45

∘

sin 45

∘

= cos45

∘

2 cos

= 2

2 cos

= 2

2 cos

= Φ

2 cos

= 3

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Weird Arithmetic

Characterizing the distribution of the planets around the sun seems to defy a mathematical

expression. Even the Titius-Bode rule falls apart pretty badly at Neptune."

The Titius-Bode Rule is:"

Which produces the orbits of the planets in astronomical units as such in AU:"

r = 0.4 + (0.3)2

n = − ∞,0,1,2,…

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However, I ﬁnd if we break-up the solar system into two parts; planets interior to the asteroid

belt, and planets exterior to the asteroid belt, quite an interesting pattern forms:!

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Thus we can deﬁne a weird arithmetic too. We begin by changing the order of operations and

say that:"

So that"

2n-3=2(2-3)=2(-1)=-2"

We see that since in normal math"

(1)x=x"

Is the multiplicative identity and"

x+0=x"

Is the additive identity that in weird math"

-1/4=4, -1/2=2,…"

Thus,…"

2(1-3)=-4=1/4=0.25=mercury=0.4 AU"

2(2-3)=-2=1/2=0.5=venus=0.72 AU"

2(3-3)=0.0=1=Earth=1.00 AU"

(In weird math zero the additive identity is the 1 the multiplicative identity. Which, resolves the

enigma of inﬁnity: which says because inﬁnity is

now 1/0=1/1=1 inﬁnity is 1, a whole encompassing any multiplicative of it.)"

2(4-3)=2=Mars=1.52 AU"

2(5-3)=4=Asteroids=2-2AU"

2(6-3)=6=Jupiter=5.2 AU"

2(7-3)=8=Saturn=9.5 AU"

2(8-3)=10=Uranus=19 AU"

2(9-3)=12=Neptune=30 AU"

Then we establish the connection between regular math and weird math by taking "

Which has plots:!

2n − 3 = 2(n − 3)

1/0 = 2/0,3/0,… = ∞

∞ = 2∞ = 3∞

2n − 3

2n − 6

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Its Taylor Expansion is:"

Which approaches the orbits of the ﬁrst ﬁve planets including the asteroid belt if we take each

term separately:"

Then after the asteroids it skips to n to the sixth for Jupiter:"

Thus, the after the asteroid the exponent of n counts 6, 9, 11,12 which is"

9-6=3 and 11-9=2 and 12-11=1 producing"

3, 2, 1"

2n − 3

2n − 6

−

162

+ O(n

)

n = 18 = 4.24264

= 0.5 = mercur y = 0.4AU

= 0.7 = venus = 0.72AU

= 1.00 = ear th = 1.00AU

= 1.412 ≈ 2 = m ars = 1.52AU

162

= 1.999 ≈ 2 = asteroids = 2AU − 3AU

1458

= 5.1 = jupiter = 5.2AU

39366

= 11.31AU = sat ur n = 9.5AU

354294

= 22.624AU = ura nus = 19AU

1062882

= 32AU = neptune = 30AU

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Thus the equation for the planets is:"

Which is"

i=(1, 2, 3, 4, 5..)"

If we write it"

i=(0, 1, 2, 3, 4…)"

x=(1, 2, 3, 4, 5..)"

The plots are…!

(

162

)

(

2 ⋅ 3

, . . .

)

2 ⋅ 3

(

i+1

2 ⋅ 3

)

0+1

2 ⋅ 3

1+1

2 ⋅ 3

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We can produce the exponents for the planets with a short piece of code. The idea that we can

model reality with a few short lines of code instead of algebra, comes from Wolfram. If you

know C++, here is the code"

// main.cpp

// planets

// Created by Ian Beardsley on 12/4/20.

#include <iostream>

int main(int argc, const char * argv[])

{

int i=0;

{

printf("0\n");

while (i!=4)

{

i=i+1;

printf("%i\n",i);

}

int i=4;

printf("\n");

while (i!=1)

{

i=i-1;

printf("%i\n",i);

}

return 0;

}

And Running the Code we have:"

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/Users/beardsleyian/Desktop/planets ; exit;

Ians-MBP:~ beardsleyian$ /Users/beardsleyian/Desktop/planets ; exit;

logout

Saving session...

...copying shared history...

...saving history...truncating history files...

...completed.

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Mathematical Look At Weird Calculus!

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Referring to my paper Weird Mathematics and The Planets (Beardsley, 2020), the weird

derivative was provoked by the planets, but I feel one should look at it mathematically. While I

will do this time permitting I would just like to present the immediate relationships between

y=f(x) and its weird derivative for f(x) equal to e to the x, and not put it in the paper until I can

go into it more extensively."

We look at the plot of e to the x whose derivative is e to the x:!

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Then we look at the plot of the weird derivative of e to the x and get:"

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We see its maximum is not to pass one and that its limit as x goes to inﬁnity is zero, and that

the graph of the e to the x and its weird derivative on the same graph speak of the graph of the

tangent function:"

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Thus we put the tangent function in the graph of both e to the x and its weird derivative and we

see it more clearly:"

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This is interesting because tangent being y/x is similar to the slope of the tangent, the

derivative (dy/dx) hence the terminology that the the derivative of y=f(x) evaluated at x is the

slope of the tangent of f(x) at (x, y). To see the meaning of this we plot by hand and make the

necessary calculations:"

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Where the plot labeled tan(x) is really tan(x)+1/2 which is what I thing we should really be

looking at. Which is pivotal if we consider what we wrote earlier:"

Let us say a/b=x, the golden ratio. Then,…"

Let us diﬀerentiate this implicitly:"

− x − 1 = 0

−

x −

1 = 0

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Which is similar to Euler’s number, e because it is the base such that is itself :"

But "

Which says for this angle the x-component equals the y-component is that is , x=1/2

bisects a right angle. Which similar in concept to Euler’s number e because it is the base such

that is itself . But if , then:"

It is the diagonal of the unit square. We notice something interesting happens:"

, , "

Where is the cosine of 30 degrees, in the unit equilateral triangle in which the altitude has

been drawn in (ﬁg 14):"

2x − 1 = 0

x =

= e

sin 45

∘

= cos45

∘

sin 45

∘

= cos45

∘

2 cos

= 2

2 cos

= 2

2 cos

= Φ

2 cos

= 3

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Silicon and Carbon

The golden ratio and the golden ratio conjugate are the solution of the quadratic"

that meets the conditions and a=b+c"

Where and , ."

We guess that artiﬁcial intelligence (AI) has the golden ratio, or its conjugate in its means

geometric, harmonic, and arithmetic by molar mass by taking these means between doping

agents phosphorus (P) and boron (B) divided by semiconductor material silicon (Si) :"

Which can be written"

We see that the biological elements, H, N, C, O compared to the AI elements P, B, Si is the

golden ratio conjugate (phi) as well:"

So we can now establish the connection between artiﬁcial intelligence and biological life:"

Which can be written:"

(

)

−

− 1 = 0

Φ =

ϕ =

5 − 1

ϕ =

(30.97)(10.81)

28.09

= 0.65

2PB

P + B

2(30.97)(10.81)

30.97 + 10.81

28.09

= 0.57

0.65 + 0.57

= 0.61 ≈ ϕ

PB(P + B) + 2PB

2(P + B)Si

≈ ϕ

C + N + O + H

P + B + Si

≈ ϕ

(P + B + Si )

PB(P + B) + 2PB

2(P + B)Si

≈ (C + N + O + H )

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Where HNCO is isocyanic acid, the most basic organic compound. We write in the arithmetic

mean:"

Which is nice because we can write in the second ﬁrst generation semiconductor as well

(germanium) and the doping agents gallium (Ga) and arsenic (As):"

Where"

Where ZnSe is zinc selenide, an intrinsic semiconductor used in AI, meaning it doesn’t require

doping agents. We now have:"

Germanium And Carbon

We could begin with semiconductor germanium (Ge) and doping agents gallium (Ga) and

Phosphorus (P) and we get a similar equation:"

, "

In grams per mole. Then we compare these molar masses to the molar masses of the

semiconductor material Ge:"

Then, take the arithmetic mean between these:"

[

+ 1

]

2PB

P + B

[

+ 1

]

≈ 2HCNO

[

PB +

2PB

P + B

][

+ 1

]

≈ 3HNCO

[

PB +

2PB

P + B

][

+ 1

]

≈ HNCO

[

+ 1

]

≈

[

+ 1

]

[

+ 1

]

(

)

2PB

P + B

(

)

P + B

(

)

≈ HNCO

2Ga P

Ga + P

= 42.866

Ga P = 46.46749

2Ga P

Ga + P

42.866

72.61

= 0.59

Ga P

46.46749

72.61

= 0.64

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We then notice this is about the golden ratio conjugate, , which is the inverse of the golden

ratio, . . Thus, we have"

1. "

2. "

This is considering the elements of artiﬁcial intelligence (AI) Ga, P, Ge, Si. Since we want to ﬁnd

the connection of artiﬁcial intelligence to biological life, we compare these to the biological

elements most abundant by mass carbon (C), hydrogen (H), nitrogen (N), oxygen (O),

phosphorus (P), sulfur (S). We write these CHNOPS (C+H+N+O+P+S) and ﬁnd:"

A similar thing can be done with germanium, Ge, and gallium, Ga, and arsenic, As, this time

using CHNOPS the most abundant biological elements by mass:"

We can also make a construct for silicon doped with gallium and phosphorus:"

0.59 + 0.64

= 0.615

ϕ ≈

Ga P(G a + P) + 2G aP

2(Ga + P)Ge

≈ ϕ

Ga P(G a + P) + 2G aP

2(Ga + P)Si

≈ Φ

CHNOPS

Ga + As + Ge

≈

[

Ga As +

2Ga As

Ga + As

][

+ 1

]

≈ CHNOPS

[

+ 1

]

Ga As

(

)

2Ga As

Ga + As

(

)

Ga + As

(

)

≈ CHNOPS

≈

[

+ 1

]

[

+ 1

]

Ga As(G a + As) + 2Ga As

2(Ga + As)Ge

≈ 1

C + H + N + O + P + S

Ga + As + Ge

≈

(C + N + O + H ) ≈

2(Ga + P)Si

Ga P(G a + P) + 2G aP

(P + B + Si )

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And for germanium doped with gallium and phosphorus:"

Now we develop a scheme for naming the AI elements with numbers that might reveal

patterns."

The Scheme

Silicon and germanium are in group 14 like carbon (C) and as such have 4 valence electrons. Thus to have

positive type silicon and germanium, they need doping agents from group 13 (three valence electrons)

like boron and gallium, and to have negative type silicon and germanium they need doping agents from

group 15 like phosphorus and arsenic.

We need a scheme that takes all of this into account.We begin with the periodic table of the

elements:"

HNCO ≈

2(Ga + P)Si

(Ga + P)

[

Ga P +

2GaP

Ga + P

]

(P + B + Si )

HNCO ≈

2(P + B + Si )Si

Ga P +

2GaP

Ga + P

Ga P(G a + P) + 2G aP

2(Ga + P)Ge

≈ ϕ

[

Ga P +

2Ga P

Ga + P

][

]

≈ HNCO

[

+ 1

]

Ga P

(

)

2Ga P

Ga + P

(

)

Ga + P

(

)

≈ HNCO

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And then pull out that section with the primary artiﬁcial intelligence (AI) elements:"

And we next number the elements as such,…"

Thus 13 is an element in row 1 and must be boron because it has three valence electrons. 14 is carbon

because it is in row 1 and has four valence electrons. And so on. So, instead of element Si we have E_24,

and instead of element Ge, we have E_34, and so on. Thus,…

Becomes,…"

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Where E means “element”. Now we deﬁne operators for the geometric, harmonic, and

arithmetic means:"

G = a b

H =

2ab

a + b

A =

a + b

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And our equation: Becomes:"

And Becomes"

The full set of AIbioequations using the standard notation is:!

PB(P + B) + 2PB

2(P + B)Si

≈ ϕ

G(E

, E

) +

H(E

, E

)

≈ ϕ

[

PB +

2PB

P + B

][

+ 1

]

≈ HNCO

[

+ 1

]

[

G(E

, E

) +

H(E

, E

) +

A(E

, E

)

]

[

+ 1

]

≈ HNCO

[

+ 1

]

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[

PB +

2PB

P + B

][

+ 1

]

≈ HNCO

[

+ 1

]

[

Ga As +

2Ga As

Ga + As

][

+ 1

]

≈ CHNOPS

[

+ 1

]

[

Ga P +

2Ga P

Ga + P

][

]

≈ HNCO

[

+ 1

]

HNCO ≈

2(P + B + Si )Si

Ga P +

2GaP

Ga + P

PB(P + B) + 2PB

2(P + B)Si

≈ ϕ

Ga As(G a + As) + 2Ga As

2(Ga + As)Ge

≈ 1

Ga P(G a + P) + 2G aP

2(Ga + P)Ge

≈ ϕ

Ga P(G a + P) + 2G aP

2(Ga + P)Si

≈ Φ

C + N + O + H

P + B + Si

≈ ϕ

C + H + N + O + P + S

Ga + As + Ge

≈

[

+ 1

]

[

+ 1

]

≈

[

+ 1

]

[

+ 1

]

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The Planets and AI!

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However, if we leave the microscopic world and go to the macroscopic, the planets, we have

the the 3/2 appears in the ratio of the terrestrial planets Earth, and Mars, which is 1.52 which I

said was in earlier work was:"

However, we used values of Si=28.09 and Ge=72.61. Recent measurements of Ge give a

slightly modiﬁed value for Ge. It is 72.64. Let us compute these two equations since they are,

as we are guessing the most important, in terms of Si, to two places after the decimal, and for

Ge using the most recent value:"

Ge=72.64 g/mol"

Si=28.0855 g/mol"

Thus,…"

We see it is still accurate to two places after the decimal. Let us now look at Mars,…"

It needs to be close to 1.52AU and we see we made a mistake with the equation to begin with.

We ﬁnd the answer is in averaging this with the diﬀerence of squares in the denominator with

the square of the diﬀerence in denominator, that is"

with "

Which gives our modiﬁed Mars equation:"

We compute its accuracy:"

This has an accuracy of:"

, 97%"

1.52 =

2SiGe

− Si

72.64

2(28.0855)(72.64) +

28.0855

72.64

1 +

28.0855

72.64

= 0.722995806

2SiGe

− Si

2(28.0855)(72.64)

72.64

− 28.0855

= 0.909

2SiGe

− Si

2SiGe

(Ge − Si)

m ars =

2SiGe

(Si − Ge)

(Si + Ge)

m ars =

2(28.0855)(72.64)

199950.5396

= 1.48AU

1.48/1.52 = 0.973684

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The plot is still like a propeller, but more stretched out. It was"

But, the modiﬁed Mars equation give us:"

Our updated AI planetary table becomes…!

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We proceed to look at what the equation for the earth might look like (1.00 astronomical units).

It is is a related polynomial that followed in my mind from the above. I simply wrote the

denominator with the square of the diﬀerences instead of the the diﬀerence of the squares, and

did not multiply the product between Si and Ge by 2. It works nearly exact as well:"

Which can be written"

and has plot"

We look at this and guess Venus is:"

Or,"

It turns out it is the arithmetic mean between the two which is:"

1.01 =

SiGe

(Ge − Si)

SiGe

(Ge

− 2SiGe + Si

)

2SiGe

+ Si

2SiGe

2Si

0.72 =

2SiGe +

1 +

of 45 88

And has plot"

All of this using Si=28.09 g/mol, and Ge=72.61g/mol!

of 46 88

Mars "

Earth "

1.48 =

2SiGe

(Si − Ge)

(Si + Ge)

1.01 =

SiGe

(Ge − Si)

of 47 88

Venus !

0.72 =

2SiGe +

1 +

of 48 88

Earth equation!

of 49 88

I said the Venus orbit was "

And, that I arrived at it by averaging my two estimates, which were"

and,…"

It is then simple to attain the equation for mercury because its average distance from the sun is

0.4 astronomical units, but since it has a highly eccentric orbit it can drop to below this or go

above it. The orbit can be described then by one half either one of these equations. Their plots

are:"

0.72 =

2SiGe +

1 +

2SiGe

+ Si

2SiGe

2Si

of 50 88

Now we move to the outer planets, most of which are gas giants. It becomes obvious what we

do for Jupiter, which is at 5.2 AU from the sun on the average. We take the earth equation:"

And, invert it, put the polynomial in the numerator, and the product in the denominator, but to

follow previous patterns, reverse the negative sign and we have:"

It is incredible how close it is to 5.2 AU, and it can be made to work perfectly by adjusting the

coeﬃcients of the polynomial, however we want as above because we want the polynomial in

the numerator to be a perfect square. That is, it can be factored"

Its plot is"

SiGe

+ Si

= 0.3

SiGe

= 0.387

SiGe

(Ge

− 2SiGe + Si

)

= 1.01

+ 2SiGe + Ge

SiGe

= 4.97AU ≈ 5AU

(Si + Ge)

SiGe

= 4.97

of 51 88

We now move out to Saturn and the solution is simple. Just as the Mercury equations were half

the Venus equations, Saturn (9 AU) is twice the Jupiter equation:"

Which of course, like the Jupiter equation can have its coeﬃcients adjusted. It plot is:"

The next planet out, Uranus (19.2 AU), is clearly about twice the Saturn Equation:"

And, Neptune (30.33 AU) is clearly three times the Saturn Equation:"

Pluto has been declassiﬁed as a planet."

2(Si + Ge)

SiGe

= 9.94

4(Si + Ge)

SiGe

= 19.88

6(Si + Ge)

SiGe

= 29.82

of 52 88

The Protoplanetary Disc

If our protoplanetary disc, as I suggested, can be taken as having a molar mass that is the

mean between germanium and silicon, then it can be taken as having a density that is the

mean between the density of germanium and silicon."

of 53 88

Taking the protoplanetary disc as a thin disc we integrate from its center to the edge, with

density decreasing linearly to zero at the edge. Thus, if the density function is given by"

And, our integral is"

The mass of the solar system adding up all the planets yields"

That accounts for"

82% of the mass of the solar system not including the sun, that is, of the

protoplanetary disc surrounding the sun."

Using germanium alone, we get,"

If we weight the mixture of silicon and germanium as 1/3 and 2/3, then we have"

Which is very close."

93%"

This is all very good, because I only used the planets and asteroids."

Si + Ge

2.33 + 5.323

= 3.8265g/cm

ρ(r) = ρ

(

1 −

)

M =

∫

2π

∫

(

1 −

)

rdrdθ

M =

πρ

π(3.8265)(7.4 × 10

)

= 2.194 × 10

grams

M = 2.668 × 10

grams

2.194

2.668

100 =

π(5.323)(7.4 × 10

)

= 3.05 × 10

grams

π(4.32467)(7.4 × 10

)

= 2.48 × 10

grams

2.48

2.668

100 =

of 54 88

Weighting silicon and germanium as 1/4 and 3/4 we have"

Which accounts for"

98%"

Of the mass of the solar system (very accurate)."

This mixture of 1/4 to 3/4 is a combination that exists in the Earth atmosphere which is

approximately the mixture of oxygen to nitrogen. The earth atmosphere can be considered a

mixture of chieﬂy O2 and N2 in these proportions:"

Air is about 25% oxygen gas (O2) by volume and 75% nitrogen gas (N2) by volume meaning

the molar mass of air as a mixture is:"

By molar mass the ratio of air to H20 (water) is about the golden ratio:"

I am not saying the solar system is a thin disk with density of the weighted mean somewhere

between silicon and germanium, but that it can be modeled as such, though if the

protoplanetary disk that eclipses epsilon aurigae every 27 years is any indication of what a

protoplanetary cloud is like, it is a thin disk in the sense that it is about 1 AU thick and 10 AU in

diameter. This around a star orbiting another star."

π(4.4.57475)(7.4 × 10

)

= 2.623 × 10

grams

2.623

2.668

100 =

0.25O

+ 0.75N

≈ air

air

≈ Φ

of 55 88

Bone!

of 56 88

Part 1: Earlier Work"

It would seem there is some possibility that as life goes from the fundamental framework to

form, so does the math that describes it. I make no attempt to understand why the

composition of life is aesthetically pleasing and how that came about, but I think it is due to the

need for function. The structure herein found arose when I was comparing biological life to

artiﬁcial intelligence. The case for bone was so interesting to me that I decided to proceed to

muscle and skin. It is because of this structure only revealing itself when comparing the

biological to AI, that I might suggest you can only speak about what life is relative to another

construct, like AI. Especially where awareness is concerned if we consider the Turing test.!

of 57 88

In my exploration of the connection between biological life and AI the most dynamic

component is that of bone. It aﬀords us the opportunity to look at:"

Multiplying Binomials"

Completing The Square"

The Quadratic Formula"

Ratios"

Proportions"

The Golden Ratio"

The Square Root of Two"

The Harmonic Mean!

of 58 88

Density of silicon is Si=2.33 grams per cubic centimeter."

Density of germanium is Ge=5.323 grams per cubic centimeter."

Density of hydroxyapatite is HA=3.00 grams per cubic centimeter."

This is"

where "

Where HA is the mineral component of bone, Si is an AI semiconductor material and Ge is an

AI semiconductor material. This means"

The harmonic mean between Si and Ge is HA,…"

This is the sextic,…"

Which has a solution"

Where x=Si, and y=Ge. It works for density and molar mass. It can be solved with the online

Wolfram Alpha computational engine. But,…"

Si +

Ge ≈ H A

HA = Ca

(PO

)

Si +

[

1 −

]

Ge = HA

2SiGe

Si + Ge

≈ HA

(x + y)

− x y(x + y)

+ 2x y

(x + y)

− 4x

(x + y)

= 0

2 + 1

−

Si +

[

− 1

]

= 0

Si =

−

4Ge

+ 4

Si = Ge − HA

of 59 88

Si +

[

1 −

]

Ge = HA

+ Ge −

Ge ≈ H A

−

Si + Ge ≈ HA

−

Si +

≈ 1

−

Si +

− 1 ≈ 0

−

Si +

[

− 1

]

= 0

of 60 88

We see that the square of the binomial is a quadratic where the third term is the square of one

half the middle coeﬃcient. This gives us a method to solve quadratics called completing the

square:"

(x + a)(x + a) = x

+ 2a x + a

(x + a)

= x

+ 2a x + a

a x

+ bx + c = 0

a x

+ bx = − c

x = −

(

)

x +

= −

(

x +

)

− 4ac

x +

− 4ac

x =

−b

− 4ac

of 61 88

−

Si +

[

− 1

]

= 0

x =

−b

− 4ac

a =

b = −

c =

[

− 1

]

− 4ac =

− 4

[

− 1

]

−

4Ge

[

−

4Ge

+ 4

]

− 4ac =

(

− 2

)

x =

[

− 2

]

[

− 2

]

Ge − H A

Si =

Ge +

Ge − H A

Si = Ge − HA

of 62 88

Si ≈ Ge − H A

HA ≈

2SiGe

Si + Ge

Si ≈ Ge −

2SiGe

Si + Ge

(Si + Ge)Ge

Si + Ge

−

(Si + Ge)Si

Si + Ge

−

2SiGe

Si + Ge

= 0

− 2SiGe − Si

Si + Ge

= 0

− 2x y − y

= 0

− 2x y = y

− 2x y + y

= 2y

(x − y)

= 2y

x − y =

x = y + 2y

x = y(1 + 2)

= 1 + 2

2 + 1

≈

2 + 1

of 63 88

A ratio is and a proportion is which means a is to b as b is to c."

The Golden Ratio "

and. "

or "

(

)

a = b + c

ac = b

c =

a = b +

− a + b = 0

− 1 +

= 0

(

)

− 1 = 0

(

)

= 1 +

(

)

= −

5 − 1

5 + 1

ϕ =

5 − 1

Φ =

5 + 1

ϕ =

of 64 88

The mineral component of bone hydroxyapatite (HA) is"

The organic component of bone is collagen which is"

We have"

(PO

)

OH = 502.32

mol

= 1298.67

mol

(PO

)

= 0.386795722

ϕ = 0.618033989

1 − ϕ = 0.381966011

(PO

)

≈ (1 − ϕ)

0.381966011

0.386795722

100 = 98.75

28.09

72.61

= 0.386861314 ≈ (1 − ϕ)

≈

(PO

)

of 65 88

If Then, for any perfect square in general…

If… x=Si; B=Ge; A=HA we have…

We make the substitutions…

; ;

To get:

If ; ;

Then,…

If x=Si, B=Ge, A=HA then…

If k=4 then we have:

Si = Ge − H A → x = B − A

[

(

)

− k

(

)

]

= b

− 4ac

[

(

)

− k

(

)

]

= b

− 4ac

a =

b =

c =

−

[

(

)

− k

(

)

]

= b

− 4ac

a =

b =

c =

−

x +

[

−

]

= 0

−

Si +

[

−

]

= 0

−

Si +

[

− 1

]

= 0

of 66 88

If we want to turn Si=Ge-HA into

We write it:

Si+HA-Ge=0 and multiply through by Si:

Then, divide through by and get:

We see that

Thus we must multiply the third term in

By That is,…

Thus we define

and And we have…

Mapping our linear equation to a quadratic.

−

Si +

[

− 1

]

= 0

+ HA ⋅ Si − Ge ⋅ Si = 0

−

Si +

= 0

[

− 1

]

−

Si +

= 0

[

−

]

[

−

]

[

− 1

]

R =

D =

[

−

]

[

x − B +

D(A)

]

= 0 →

−

x +

[

− 1

]

= 0

of 67 88

Boron

Boron is a doping agent for artiﬁcial intelligence semiconductor

materials, which are primarily silicon and germanium

But, it stands in relation on the periodic table diﬀerently than the other

principle doping agents, phosphorus, arsenic and gallium. This results

in some interesting equations.!

of 68 88

Periodic Table of the AI Elements

It is quite interesting that we can pull out the core artiﬁcial intelligence elements from the

periodic table of the elements in such a way that it allows for a three by three matrix:"

Because we can number them according to their properties:"

13 is an element in row 1 and must be boron because it has three valence electrons. 14 is carbon because

it is in row 1 and has four valence electrons. And so on. So, instead of element Si we have E_24, and

instead of element Ge, we have E_34, and so on. B and P and Ga and As are doping agents for Si and Ge.

Three by three matrixes are the most important because they are associated with vectors in three

dimensional space, and thus are used for taking the cross product between two vectors.

00001: The properties of artiﬁcial intelligence substances form the basis of Universal Structure.

of 69 88

Asymmetry in AI Elements

The primary elements of artiﬁcial intelligence (AI) used to make diodes and transistors, silicon (Si) and

germanium (Ge) doped with boron (B) and phosphorus (P) or gallium (Ga) and arsenic (As) have an

asymmetry due to boron. Silicon and germanium are in group 14 like carbon (C) and as such have 4

valence electrons. Thus to have positive type silicon and germanium, they need doping agents from group

13 (three valence electrons) like boron and gallium, and to have negative type silicon and germanium they

need doping agents from group 15 like phosphorus and arsenic. But where gallium and arsenic are in the

same period as germanium, boron is in a different period than silicon (period 2) while phosphorus is not

(period 3). Thus aluminum (Al) is in boron’s place. This results in an interesting equation.

The differential across germanium crossed with silicon plus the differential across silicon crossed with

germanium normalized by the product between silicon and germanium is equal to the boron divided by

the average between the germanium and the silicon. The equation has nearly 100% accuracy:

We found (Beardsley, Mathematical Structure, 2020) that the diﬀerential across silicon (P-Al)

times germanium (Ge) over boron (B) plus the diﬀerential across germanium (As-Ga) times

silicon (Si) over boron (B) was equal to the harmonic mean between Si and Ge. This was

interesting because aluminum is used as what I called a dummy doping agent element, which

when inserted predicts the actually doping agent boron, that seems out of place in the periodic

table where the core artiﬁcial intelligence elements are concerned. This is written:"

I ﬁnd it interesting that the AI elements can be pulled out in a 3X3 matrix because the AI elements seem

to be core to everything in the Universe. I wrote earlier:

00002: It may be we can only understand biological life relative to some other construct, like artiﬁcial

intelligence.

Si(A s − Ga) + Ge(P − Al )

SiG e

G e + Si

28.09(74.92 − 69.72) + 72.61(30.97 − 26.98)

(28.09)(72.61)

2(10.81)

(72.61 + 28.09)

0.213658912 = 0.21469712

0.213658912

0.21469712

= 0.995

(As − G a) +

G e

(P − Al ) =

2 SiGe

Si + Ge

of 70 88

Stokes Theorem states:

We know the harmonic mean H of a function is

And, that the arithmetic mean A of a function is

We have

∫

( ∇ ×

⇀

u ) ⋅ d

⇀

S =

∮

⇀

u ⋅ d

⇀

∇ ×

⇀

u =

⇀

⃗

⇀

∂

∂x

∂

∂y

∂

∂z

⃗

∂

∂x

∂

∂y

∂

∂z

(Ga)z

(As)y

(As − G a)

⃗

∂

∂x

∂

∂y

∂

∂z

(Al )z 0

(P)x

(P − Al )

⃗

u = (u

, u

)

⃗

v = (v

, v

)

⃗

u = 0

⃗

i +

(Ga)z

⃗

j +

(As)y

⃗

u = 0

⃗

i +

(Ga)z

⃗

j +

(As)y

⃗

H =

b − a

∫

f (x)

−1

d x

A =

b − a

∫

f (x)d x

(As − G a) +

(P − Al ) =

Ge − Si

∫

of 71 88

But, we want to use Stokes theorem so we want the integral in the numerator. So, we make the

approximation

And, we have

But, this is only 80% accurate. We ﬁnd it is very accurate if we say

Which yields

We have by molar mass

Thus,…

H ≈ A

∫

[

(As − G a) +

(P − Al )

]

d x d y ≈

Ge − Si

∫

x d x

f (x) =

∫

[

(As − G a) +

(P − Al )

]

d x d y ≈

Ge − Si

∫

x d x

(Ga) =

28.09

10.81

(69.72) = 181.1688g/mol

(Al ) =

72.61

10,81

(26.98) = 181.2227g/mol

(As) =

28.09

10.81

(74.92) = 194.68111g/mol

72.61

10.81

(30.97) = 208.02328g/mol

⃗

u = 181z

⃗

j + 195y

⃗

v = 181z

⃗

i + 208y

⃗

of 72 88

We can break up our integral into two integrals u, and v:

Holding x constant and y constant in our u and v integrals we have subtle eclipses in the y-z

planes and x-z planes respectively:

∫

(As − G a)d yd z ≈

(Ge − Si)

∫

x d x

∫

(P − Al )d x dz ≈

(Ge − Si)

∫

yd y

of 73 88

By Molar Mass

We found that the diﬀerential across silicon (P-Al) times germanium (Ge) over boron (B) plus the

diﬀerential across germanium (As-Ga) times silicon (Si) over boron (B) was equal to the

harmonic mean between Si and Ge. This was interesting because aluminum is used as what I

called a dummy doping agent element, which when inserted predicts the actually doping agent

boron, that seems out of place in the periodic table where the core artiﬁcial intelligence

elements are concerned. This is written:"

(As − G a) +

(P − Al ) =

2SiG e

Si + Ge

of 74 88

By Density

We ask if the asymmetry in the AI elements in the periodic table due to boron results in a

dynamic equation by molar mass, then does it as well by density? While molar mass is due to the

composition of elements, density is due to the balance between the strong nuclear force holding

protons together balanced by their electric forces that are mutually repulsive.

The density of boron is 2.340 grams per cubic centimeter, that of phosphorus (white

phosphorus) is 1.88 grams per cubic centimeter and gallium is 5.904 grams per cubic centimeter.

of 75 88

Arsenic is 5.7 grams per cubic centimeter, germanium is 5.323 grams per cubic centimeter and

aluminum is 2.7 grams per cubic centimeter. We have the following scenario:

Again we see boron breaks the symmetry in that period three densities are on the same order and

period 4 densities are on the same order, but that of boron is almost the same as silicon in period

three. We see that semiconductor material Si is the the average between doping agent P and

would be doping agent aluminum that takes the place of boron and, that, the average doping

agent Ga and semiconductor material Ge is approximately the average of doping agent As. Thus

we have:

Ga-As=0.204 (differential across Ge)

Al-P=0.82 (dummy differential across Si)

Si~B

Ge/Si~B

(0.82)

= 0.816

(0.204)

= 0.464

0.816

0.464

= 1.7586

GeSi = 3.52

3.52

= 1.76 ≈ 1.7586

of 76 88

And, we have

The ﬁrst factor takes the form of the harmonic mean between a and b, H:

And the second term takes the form of the geometric mean between a and b, G:

The equation is 94.68% accurate.

Aluminum, while a dummy in the equation used to arrive at the dynamics due to asymmetry by

way of boron actually is widely used in AI because it is a conductor, which makes it an electric

shield, so it can be used to enclose electrical circuitry to protect it from electric ﬁelds. Thus we

have the two equations by molar mass and density respectively:

But

Can be written

(Al − P )

Ga − As

B ≈

GeSi

2 ≈

Si + P

B ≈

2Ge(Ga − As)

Si(Al − P) + P(Al − P)

GeSi

H =

2a b

a + b

G = a b

2(5.323)(5.904 − 5.7)

2.33(2.7 − 1.88) + 1.88(2.7 − 1.88)

(5.323)(2.33) = 2.2155

2.2155

2.340

= 94.68

B ≈

2Ge(Ga − As)

Si(Al − P) + P(Al − P)

GeSi

∫

[

(As − G a) +

(P − Al )

]

d x d y ≈

Ge − Si

∫

x d x

B ≈

2Ge(Ga − As)

Si(Al − P) + P(Al − P)

GeSi

of 77 88

But

Is the geometric mean between Ge and Si. The geometric mean between a and b is given by:

Thus our equation in terms of density can be put in integral form as well:

And we see that this integral is correct:

Which is close to

94.5%

2Ge(Ga − As)

[

Si(Al − P) + P(Al − P)

]

≈ GeSi

GeSi

f = ex p

(

b − a

∫

log f (x)d x

)

∫

[

2Ge(Ga − As)

[

Si(Al − P) + P(Al − P)

]

d x d y = ex p

(

Ge − Si

∫

log(x)d x

)

5.323(ln(5.323) − 5.323) − 2.33(ln(2.33) − 2.33) = 3.936

3.936

5.323 − 2.33

= 1.315

1.315

= 3.725

(5.323)(2.33) = 3.52

3.52

3.725

= 0.945

of 78 88

By Atomic Radius

I then considered atomic radii of these elements.

of 79 88

of 80 88

The atomic radii data varies some from source to source. Here I use data set 4 (previous page)

that is the average of respective values in data sets 1, 2, and 3. We have looked at molar mass,

and density and the dynamic relationships that result in terms of them do to the asymmetry

introduced into the AI elements in the periodic table. It is natural to look at radius next. It has a

lot to do with the structure and properties of the elements just like is true of the their molar

masses and densities.

Here we have the differential across Si, times Si/B plus the differential across Ge times Ge/B is

the golden ratio, phi, times the arithmetic mean between Si and Ge in atomic radius.

(Al-P)=143-116=36

(Ga-As)=19

(36)(115/88)=36(1.3)=46.8

19(123/88)=19(1.4)=26.6

46.8+26.6=73.4

(115+123)/2=119

The golden ratio and the golden ratio conjugate are the solution of the quadratic

that meets the conditions and a=b+c"

Where and , .

We have already said

Thus by radius the integral form of the equation is:

119

73.4

= 1.62 ≈ Φ = 1.618

(Al − P )

+ (G a − As)

= Φ

Si + Ge

(

)

−

− 1 = 0

Φ =

ϕ =

5 − 1

ϕ =

f =

b − a

∫

f (x)d x

of 81 88

∫

[

(Al − P )

+ (G a − As)

]

d x d y =

Ge − Si

∫

x d x

of 82 88

The Generalized Equation

Returning to the Asymmetry in the AI Periodic Table:

and by molar mass or

and by density or

and by atomic radius

Are respectively and

And, the ratios

and by molar mass or

and by density or

and by atomic radius

Are, quotients , and , respectively, then if

And, since the geometric mean, arithmetic mean, quadratic mean (root mean square), harmonic

mean are special cases of the generalized mean:

p= 0,1, 2, -1,…

Then the generalized form of our equations is:

Where C is some constant. This is all the following equations:

(As − G a)

(P − Al )

(Al − P )

(Ga − As)

ΔE

2Ge(Ga − As)

∇ = (ΔE

, ΔE

)

Q = (Q

, Q

)

(

∑

i=1

)

∇ ⋅ Q = C

(

∑

i=1

)

of 83 88

By molar mass.

By density.

By atomic radius.

While this is the arithmetic mean when p=1, and the harmonic mean for p=-1, I have used it as

the geometric mean for p=0, which we can see explodes to inﬁnity at zero. But, since I am a

physicist and not a mathematician, I assume taking the limit as p—->0 is the same as evaluating

it at zero. But, as physicists undergo courses of mathematics, I do understand how to present

equations in terms of the formalities of mathematics. The proper way to treat this as a

mathematician is, the limit as p goes to zero is the geometric mean because

However we can generalize the equation to include the geometric mean without having to take

the limit as p goes to zero using the f-mean. We have:

The power mean is obtained by letting

Thus, our equation becomes:

It is the geometric mean if

(As − G a) +

(P − Al ) =

Ge − Si

∫

[

2Ge(Ga − As)

[

Si(Al − P) + P(Al − P)

]

d x d y = ex p

(

Ge − Si

∫

log(x)d x

)

∫

[

(Al − P )

+ (G a − As)

]

d x d y =

Ge − Si

∫

x d x

, …, x

) =

∏

i=1

, …x

) = f

−1

(

⋅

∑

i=1

f (x

)

f (x) = x

∇ ⋅ Q = Cf

−1

(

⋅

∑

i=1

f (x

)

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00001: The properties of artiﬁcial intelligence substances form the basis of Universal Structure.

00002: It may be we can only understand biological life relative to some other construct, like artiﬁcial

intelligence.

00003: It is a purpose of biological life (C, N, O, H) to discover the properties of (P, B, Si) so it can make

computing machines which are ultimately necessary to its survival.

00004 That node manufacturing technology (semiconductors) is possible on this planet by way of

naturally occurring substances on this planet (Si, P, B) then the possibility exists that these elements have

such properties by intention outside of ourselves.

00005 That the semiconducting elements found on this planet have the same properties throughout the

Universe, and are in great abundance, would indicate they were not made expressly for humans.

00006: The problem is we don’t know what exists in the microcosmos, we must look at it with itself, we

must use the photon to look at the atom, and it is on the same order of magnitude, so it has to much of an

effect on it for determinism, hence the uncertainty principle saying we can’t know what is there, only the

probability of what is in a given region for a given time.

00007: I can see two scenarios here: One, since there are billions of galaxies, each with billions of stars

the probability for life to occur around a handful of them was high, and the Sun is one of these stars. Or

two, The algorithm put in place was for the Big Bang to create hydrogen, then that condensed into stars

which made from it carbon throughout the universe, and having four valence electrons, it could form with

the hydrogen in long chains, the hydrocarbons, from which DNA synthesized proteins, and life arose

across the universe as a single uniform idea throughout. Elements like silicon have four valence electrons

as well but, are more suitable for making artiﬁcial intelligence when doped with phosphorus and boron!

f (x) = log(x)

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Masculine And Feminine!

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The female sex hormone estrogen (E) is C18H24O2=272.38 g/mol and the male sex hormone

testosterone (T) is C19H28O2=288.42 g/mol. Both are made from cholesterol (Ch)

C27H46O=386.65 g/mol. If we notice that (Ch+T)/E=2.5 and Ge/Si=2.6 we write

(Ch+T)/E=Ge/Si

Results in

T(1-Si/Ge)+E(1-Ge/Si)=Ch(Si/Ge-1)

I interpreted this as the masculine (T) is in inverse relation to the feminine (E), but that the two

add up to a whole (Ch) in that the masculine has coefficient 1-Si/Ge and the feminine has

coefficient 1-Ge/Si that is they are inverse relation but compliment one another. How would an

AI use this information to determine its sex?…

The male is reduced less in the difference between 1 and Si/Ge, but the the female is reduced

less by having Ge in the numerator. It is really quite egalitarian.

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Conclusion: The question remains, what is a deﬁnition for the weird derivative in general, not

just for e to the x?!

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The Author"